进一步地，因f”(p)≠0，运用隐函数存在定理从 x=-f'(p) 解出 p=ω(x 则Clairaut方程的特解可写成 y=xo(x)+f(@(x)). (4) 思考：Clairaut方程在常微分方程基础理论中占有重要的地位。 ●Clairaut方程的特解与通解的关系如何？ 4口6+4之·4生+2风0 张样：上涛交通大学数学系 第七讲、一阶隐式微分方程-2、高阶教分方程与Mathematica求解
?ò⁄/, œ f 00(p) 6= 0, $^¤ºÍ3½nl x = −f 0 (p) )— p = ω(x). K Clairaut êßA)å§ y = xω(x) +f(ω(x)). (4) g: Clairaut êß3~á©ê߃:nÿ•”ká/†" Clairaut êßA)Üœ)'XX¤º ‹å: ˛°œåÆÍÆX 1‘˘!ò ¤™á©êß-2!p á©êßÜMathematica¶)
Clairaut方程的特解与通解的关系如下： ●过Clairaut方程特解上任一点的切线是通解中的一条直线. 事实上，设(0，y(xo)是特解(4)上的任一点，则 有y(xo)=o(xo.所以特解过(x0,y(x0)的切线方程为 y-y(xo)=o(xo)(x-xo), 即 y=cox+f(co), c0=0(x0): 。特解(4)不能用通解表示，因为o(x)不是常数.事实上， 从x+f'(o(x)三0得 1 o国=o≠0. ●上述两点说明Clairaut方程特解对应的积分曲线上任一点都 有通解中的一条积分曲线通过，且两者在该点相切， 张样：上寿交通大学数学系第七讲、一阶隐式微分方程2、高阶微分方程与Mathematica求解
Clairaut êßA)Üœ)'XXeµ L Clairaut êßA)˛?ò:ÉÇ¥œ)•ò^ÜÇ. Ø¢˛, (x0, y(x0)) ¥A) (4) ˛?ò:, K k y 0 (x0) = ω(x0). §±A)L (x0, y(x0)) ÉÇêßè y−y(x0) = ω(x0)(x−x0), = y = c0x+f(c0), c0 = ω(x0). A) (4) ÿU^œ)L´, œè ω(x) ÿ¥~Í. Ø¢˛, l x+f 0 (ω(x)) ≡ 0 ω 0 (x) = − 1 f 00(ω(x)) 6= 0. ˛„¸:`² Clairaut êßA)ÈA»©Ç˛?ò:— kœ)•ò^»©ÇœL, Ö¸ˆ3T:ÉÉ. ‹å: ˛°œåÆÍÆX 1‘˘!ò ¤™á©êß-2!p á©êßÜMathematica¶)
Clairaut方程是法国数学家Alexis Clairaut于1734年引入的. Alexis Clairaut(3 May 1713-17 May 1765)was a prominent French mathematician,astronomer,and geophysicist. o His paper procured his admission into the French Academy of Sciences in 1731,although he was below the legal age as he was only eighteen. o He was elected a Fellow of the Royal Society of London in 1737. 特别是在天文学的研究方面做出了杰出的贡献： o He obtained an ingenious approximate solution of the problem of the three bodies o In 1750 he gained the prize of theSt Petersburg Academy for his essay Theorie de la lune ●ln1759 he calculated the perihelion(近日点)of Halley's comet. 2月风0 张样：上涛交通大学数学系 第七讲、一阶隐式微分方程-2、高阶教分方程与Mathematica求解
Clairaut êߥ{IÍÆ[ Alexis Clairaut u 1734 c⁄\. Alexis Clairaut (3 May 1713–17 May 1765) was a prominent French mathematician, astronomer, and geophysicist. His paper procured his admission into the French Academy of Sciences in 1731, although he was below the legal age as he was only eighteen. He was elected a Fellow of the Royal Society of London in 1737. AO¥3U©ÆÔƒê°â— #—zµ He obtained an ingenious approximate solution of the problem of the three bodies In 1750 he gained the prize of theSt Petersburg Academy for his essay ThWorie de la lune In 1759 he calculated the perihelion(CF:) of Halley’s comet. and so on Clairaut seek to make his pleasures into his day to day work, he lost rest, health, and finally life at the age of fifty-two." ‹å: ˛°œåÆÍÆX 1‘˘!ò ¤™á©êß-2!p á©êßÜMathematica¶)
讨论+思考：在Clairaut方程中 通过f(x)的适当选取可以得到特解的哪些可能的几何形状？ 口1回1怎：“主12月双0 张样：上将交通大学数学系第七讲、一阶隐式微分方程2、高阶微分方程与Mathematica求解
?ÿ+g: 3 Clairaut êß• œL f(x) ·¿å±A)= åUA¤/G? ‹å: ˛°œåÆÍÆX 1‘˘!ò ¤™á©êß-2!p á©êßÜMathematica¶)
例2.求解方程 =-p+式 p=y(x) 解：对上述方程两边关于x求导得 p-()=0 利用该方程的特解和通解得到原方程的特解和通解为 y=,y=+c+2 其中c是任意常数. 观察与思考： ·例2中的特解与通解的关系如何？ 口0·4之·4生+2刀a0 张样：上涛交通大学数学系 第七讲、一阶隐式微分方程-2、高阶微分方程与Mathematica求解
~ 2. ¶)êß y = p 2 −xp+ 1 2 x 2 , p = y 0 (x). ): È˛„ê߸>'u x ¶ (2p−x) 1− dp dx = 0. |^TêßA)⁄œ)êßA)⁄œ)è y = 1 4 x 2 , y = 1 2 x 2 +cx+c 2 , Ÿ• c ¥?ø~Í. * Üg: ~ 2 •A)Üœ)'XX¤º ‹å: ˛°œåÆÍÆX 1‘˘!ò ¤™á©êß-2!p á©êßÜMathematica¶)